Editing Lattice (order) (section)

=== As algebraic structure ===
A '''lattice''' is an [[algebraic structure]] <math>(L, \vee, \wedge)</math>, consisting of a set <math>L</math> and two binary, commutative and associative [[Operation (mathematics)|operations]] <math>\vee</math> and <math>\wedge</math> on <math>L</math> satisfying the following axiomatic identities for all elements <math>a, b \in L</math> (sometimes called {{em|absorption laws}}):
<math display=block>a \vee (a \wedge b) = a</math>
<math display=block>a \wedge (a \vee b) = a</math>

The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.<ref>{{harvnb|Birkhoff|1948|p=[https://archive.org/details/in.ernet.dli.2015.166886/page/n35/mode/2up 18]}}. "since <math>a = a \vee (a \wedge (a \vee a)) = a \vee a</math> and dually". Birkhoff attributes this to {{harvnb|Dedekind|1897|p=[https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf#page=10 8]}}</ref> These are called {{em|idempotent laws}}.
<math display=block>a \vee a = a</math>
<math display=block>a \wedge a = a</math>

These axioms assert that both <math>(L, \vee)</math> and <math>(L, \wedge)</math> are [[semilattice]]s. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the [[Duality (order theory)|dual]] of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same [[partial order]].